1. Theta-positive representations: In two articles, François Labourie and co-authors demonstrated that positive representations form a connected component of character varieties and are Anosov, thereby discrete and faithful. This result is highly anticipated as it proves that Theta-positive representations merit special study. The concept of positive representations was utilised throughout the project.
2. Curves in the boundary of AdS and Positives Curves: The team has obtained several results in this area. Firstly, Rym Smaï and Enrico Trebeschi (with co-authors) extended in a submitted article the analysis by Christopher Bishop in the Euclidean case for $\mathbf{H}^3$ to this Lorentzian setting for the 3-dimensional Anti-de-Sitter Space (AdS). They showed the equivalence between the following facts: the curve is a graph of a Weyl–Petersson homeomorphism and the renormalized area of the corresponding maximal surfaces, whose existence is proved by François Labourie and Jérémy Toulisse (with co-author), is finite. In another article, François Labourie and Jérémy Toulisse (with co-author) showed that the Euclidean concepts of W-volumes and Epstein surfaces extend to the Lorentzian setting, giving in particular a notion of Liouville action for Lorentzian Annuli. This leads to a new invariant for positive curves which is finite for piecewise positive circles (a novel concept).
3. Functions on Anosov moduli spaces. A key component of the project was achieved by François Labourie and a coauthor. They exhibited a special class of functions on the deformation space of Anosov representations. The fundamental property of this class is that it is closed under the Poisson bracket and that this Poisson bracket can be computed combinatorially. This has resulted in several advances in the general project, which are currently at the stage of unpublished drafts.
4. Moduli spaces of Higgs bundles in family. Recently Jérémy Toulisse and colleagues explored the geometry of the moduli space of Higgs bundles in family. This occurs when the complex structure of the base surface changes. They demonstrated that this moduli space possesses a pseudo-Kähler geometry on a certain open, dense subset. Consequently, the moduli space of the space of minimal surfaces becomes a complex submanifold and remains pseudo-Kähler when restricted to cyclic representations. This represents a significant advancement in our understanding of minimal surfaces within locally symmetric spaces.
5. Minimal surfaces in high-dimensional spheres. François Labourie and Jérémy Toulisse, with local coauthors, proved the existence of a sequence of compact minimal surfaces in higher-dimensional spheres. This sequence converges to a surface with constant negative curvature, thus solving a conjecture by S.T.~Yau. This unexpected direction offers new perspectives, particularly in studying surface group representations within infinite-dimensional hyperbolic spaces.
6. Dynamical studies of Anosov representation. In a seminal preprint, W.~Thurston defined stretch maps and the related Thurston asymmetric metric as well as unit balls in the tangent space of Teichmüller spaces. These objects have been defined in the context of higher rank Lie groups. Xian Dai, with co-authors, studied the extension of stretch maps and obtained results (with another co-author) on maximal currents, where maximal is with respect to some ergodic optimisation.
7. Special submanifolds in symmetric spaces. The team investigated several types of problems. Firstly, the existence of maximal surfaces with prescribed boundary at infinity was obtained by François Labourie, Jérémy Toulisse and a co-author. Subsequently Jérémy Toulisse studied with a colleague holomorphic surfaces in the pseudo-hyperbolic space of dimension 6 and their relationship with the split form of the group $\mathbf{G}_2$ [1]. Enrico Trebeschi and a co-author also studied the scalar curvature of space-like maximal submanifolds in pseudo-hyperbolic spaces and proved it is negative.
8. Surface subgroups in lattices of Lie groups. François Labourie and co-authors prove the existence of surface subgroups in many lattices, thus generalising a famous result in 3-dimensional geometry. While the results of this paper were primarily obtained prior to the project’s commencement, this has sustained new activities. François Labourie with a coauthor is now w studying surface subgroups in other lattices, which would lead to significant advancements for the project.
